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Matrix Inversion XNA

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I have two functions for setting rendering transformations in my scene graph that are called before every scene node is rendered:

 public virtual void PreRender(Scene scene)
            // push a copy of the top matrix onto the stack
            scene.MatrixStack.Push ();
            // multiply that matrix with the one that
            // is already there and replace it with the result
            scene.MatrixStack.MultMatrix ( this.sceneProps.ParentSpaceMatrix );
            Matrix WorldTrans = scene.MatrixStack.Peek();

            //translate world space into screen space
            Matrix ScreenMat = scene.Camera.TransformToScreen(WorldTrans);

            // now this node is the root of whatever is drawn after it
            scene.Transform *= ScreenMat;

        public virtual void PostRender(Scene scene)
            // Pop the top matrix off of the stack and get us back up the tree
            scene.MatrixStack.Pop ();
            // ***Now here is the trouble spot***
            // I need to undo the transformation I just did in prerender
            // so I invert the matrix
            Matrix TransMat = Matrix.Invert ( scene.Transform );


I would think that finding the inverse of a matrix each time I rendered a scene node would make my game incredibly slow. I have heard that it could be done quickly for transformation matrices by swapping their rows and collumns or something like that. My questions are: Does XNA take advantage of what I just mentioned and make finding the inverse of a transformation matrix speedy? Is this the right way to go about un doing transformations?

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